= Solution
A convenient <root basis> is
$$
\boxed{\alpha_1=\varepsilon_1-\varepsilon_2,\qquad
\alpha_2=\varepsilon_2-\varepsilon_3,\qquad
\alpha_3=2\varepsilon_3.}
$$
The corresponding <Cartan matrix> is
$$
\left(\langle\alpha_i,\alpha_j^\vee\rangle\right)_{ij}
=\begin{pmatrix}
2&-1&0\\
-1&2&-1\\
0&-2&2
\end{pmatrix}.
$$
Thus the <Dynkin diagram> is the three-node $C_3$ chain, with a double edge between $\alpha_2$ and $\alpha_3$ and its arrow pointing toward the shorter root $\alpha_2$.
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